Preheat oven to 350 degrees F. Line a shallow baking pan with foil; set aside. Cut cheese block into six 3x1/2x1/2-inch sticks. Wrap a slice of ham around each cheese stick. Using a sharp knife, cut a pocket in each chicken breast half by cutting horizontally through the thickest portion to, but not through, the other side. Stuff a ham-wrapped cheese stick into each pocket. Secure pockets with wooden toothpicks. Sprinkle chicken with salt and pepper.
Place 1/3 cup flour in a shallow dish. In a second shallow dish combine eggs and the water. Place panko in a third shallow dish. Coat chicken with flour. Dip chicken into egg mixture, then into panko, turning to coat.
In a 12-inch skillet heat 2 tablespoons of the oil over medium heat. Add half of the chicken; cook about 4 minutes or until golden, turning once. Transfer chicken to the prepared baking pan. Repeat with the remaining oil and chicken. Bake, uncovered, for 40 to 50 minutes or until chicken is no longer pink (170 degrees F). Remove toothpicks before serving.